Classical Yang–Mills theory is scale invariant: its action contains not a single parameter of mass dimension. The quantum theory nevertheless generates a mass scale — this is dimensional transmutation, and it is the origin of the mass gap. This paper writes out the standing of the numbers that appear in that mechanism.
First, Λ_QCD is an invariant built from two conventions. The renormalisation scale μ is a convention, and the coupling g(μ) depends on it. Yet their combination Λ = μ exp(−8π²/(b₀g²)) depends on neither. This is confirmed numerically here: moving μ from 91.19 GeV to 10, 2 and 1 GeV leaves Λ unchanged at 245.75 MeV. Paper 35 separated the two 4π's inside 4πG into a fact (Gauss's flux) and a convention (a choice of units); this paper is the converse case. Convention × convention can make an invariant. "It contains a convention, therefore it has no content" does not follow — the work is to check what does not depend on the convention.
Second, there are at least three distinct roots by which a scale gets fixed. (a) Dimensional analysis — a power law whose exponents come from the inverse of the dimension matrix (the Planck length of Paper 39 has |det|=2, hence a square root; the Kolmogorov length of Paper 03 has |det|=4, hence a fourth root). (b) A choice of units — a convention, whose value may be chosen freely (Papers 35, 38). (c) Dimensional transmutation — e^(−1/g²), which carries an essential singularity at g→0 and therefore vanishes at every order of the perturbative expansion. Dimensional transmutation thus never appears, however far the perturbative calculation is carried. The three must not be lumped together as "ways a scale is determined".
Third, this number is exponentially fragile. Since d lnΛ/dα_s = 2π/(b₀ α_s²) ≈ 50, a 1% error in α_s becomes 6.1% in Λ, and 5% becomes 34%. Changing n_f from 3 to 5 moves Λ from 246 MeV to 88 MeV (153 MeV at n_f=4). This is the opposite of Paper 04, whose n* moved by only 1.3 when its input was changed by a factor of 2.5. When the quantity is exponential, logarithmic measures compress the uncertainty of the inputs, but Λ is the value of an exponential itself and so receives that uncertainty magnified. Logarithmically robust numbers and exponentially fragile ones coexist in one series. Whenever a value of Λ_QCD is quoted, the scheme, n_f and loop order must be quoted with it.
Fourth, the mass gap is a fact and Λ_QCD is bookkeeping. The lightest scalar glueball on the lattice, m(0⁺⁺)≈1.7 GeV, is a measurable physical quantity, whereas Λ_QCD depends on scheme, n_f and loop order. The ratio m(0⁺⁺)/Λ ≈ 6.9 is therefore also scheme-dependent and is not a meaningful dimensionless quantity — the discipline of Paper 38, that dimensionless ratios are the physics, applies only when both members of the ratio are physical quantities. And the Λ_QCD ≈ 246 MeV of this paper shares a numerical value with the Higgs vacuum expectation value v ≈ 246 GeV of Paper 09. This is a complete accident and implies nothing whatsoever: they differ in origin, in units and in order of magnitude, and the very number that coincides is not firm, moving down to 88 with n_f and scheme. The discipline this series has written repeatedly about 4π and 16π² (Papers 48, 86, 100, 115) is here applied explicitly to 246.
Fifth, the one-loop Λ sits where one loop breaks down. Λ is defined as the pole of the running coupling, but near it α_s exceeds one (0.333 at μ=2 GeV, 0.497 at 1 GeV, 0.983 at 0.5 GeV, 3.50 at 0.3 GeV). The definition lies outside its own domain of validity. This is not an error but the character of Λ; in the vocabulary of Paper 113 it is a module boundary — the place where α_s crosses one.
Finally, rewriting the exponent of dimensional transmutation with the loop expansion parameter α_eff = g²/16π² gives Λ = μ exp(−1/(2b₀ α_eff)), the same functional form as the instanton weight exp(−1/(2α_eff)) of Paper 45. The 16π² appearing here is the one-loop geometric measure of Paper 46, and one of the addresses in the roll call of 16π² set out in Paper 09 (3rd edition). This agreement, however, holds within known quantum field theory and was not derived here. The existence and mass gap of Yang–Mills theory is a Clay Millennium Prize Problem and is unsolved. This paper claims neither its proof, nor the construction of four-dimensional Yang–Mills theory, nor any partial achievement of either.
(scope note) Positioning of this paper: This paper offers a structural interpretation of established physical laws and mathematical theorems. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are transformations of known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). The Yang–Mills Mass Gap and Dimensional Transmutation [Paper 5]. Zenodo.
DOI: 10.5281/zenodo.21404236
ORCID: 0009-0006-8587-3450